Unraveling The Standard Form Of (1+7i)(3-i) A Comprehensive Guide
In the captivating world of mathematics, complex numbers stand as a testament to the boundless nature of mathematical exploration. These numbers, extending beyond the familiar realm of real numbers, introduce the concept of an imaginary unit, denoted by i, where i² = -1. Complex numbers, represented in the standard form of a + bi, where a and b are real numbers, open doors to a rich tapestry of mathematical concepts and applications. Within this domain, the expression (1 + 7i)(3 - i) presents an intriguing challenge, inviting us to delve into the intricacies of complex number arithmetic and unravel its standard form.
Understanding Complex Number Arithmetic
Complex number arithmetic, while seemingly daunting at first, follows a set of well-defined rules that govern their operations. Addition, subtraction, multiplication, and division of complex numbers can be performed with precision, unveiling the elegant structure that underlies these mathematical entities. To embark on our journey of simplifying (1 + 7i)(3 - i), we must first grasp the fundamentals of complex number multiplication.
Multiplication: A Gateway to Simplification
Multiplying complex numbers involves a process akin to multiplying binomials, employing the distributive property to expand the expression. When multiplying two complex numbers, (a + bi) and (c + di), we diligently apply the distributive property, ensuring that each term in the first complex number is multiplied by each term in the second complex number. This process unfolds as follows:
(a + bi)(c + di) = a(c + di) + bi(c + di)
Expanding further, we obtain:
ac + adi + bci + bdi²
Here, the term bdi² holds the key to simplification. Since i² = -1, we can substitute -1 for i², transforming the expression into:
ac + adi + bci - bd
Rearranging the terms, we arrive at the standard form of the product:
(ac - bd) + (ad + bc)i
This standard form, a + bi, elegantly separates the real and imaginary components of the complex number, providing a clear representation of its structure.
Deciphering (1 + 7i)(3 - i): A Step-by-Step Approach
Now, armed with the knowledge of complex number multiplication, we can confidently tackle the expression (1 + 7i)(3 - i). Following the distributive property, we meticulously expand the expression:
(1 + 7i)(3 - i) = 1(3 - i) + 7i(3 - i)
Expanding further, we get:
3 - i + 21i - 7i²
Substituting i² = -1, we obtain:
3 - i + 21i + 7
Combining the real and imaginary terms, we arrive at:
(3 + 7) + (-1 + 21)i
Simplifying, we unveil the standard form of the expression:
10 + 20i
Thus, the standard form of (1 + 7i)(3 - i) is 10 + 20i, where the real component is 10 and the imaginary component is 20.
The standard form of a complex number, a + bi, serves as the cornerstone for performing various operations on complex numbers. Addition, subtraction, and equality comparisons become straightforward when complex numbers are expressed in this form. The real and imaginary components can be treated as separate entities, allowing for seamless calculations.
Addition and Subtraction: A Symphony of Components
Adding or subtracting complex numbers in standard form involves a simple yet elegant process: we add or subtract the real components separately and the imaginary components separately. Consider two complex numbers, a + bi and c + di. Their sum is given by:
(a + bi) + (c + di) = (a + c) + (b + d)i
Similarly, their difference is given by:
(a + bi) - (c + di) = (a - c) + (b - d)i
These operations highlight the harmonious interplay between the real and imaginary components, showcasing the elegance of complex number arithmetic.
Equality: A Dance of Components
Two complex numbers, a + bi and c + di, are deemed equal if and only if their real components are equal and their imaginary components are equal. This fundamental principle can be expressed as:
a + bi = c + di if and only if a = c and b = d
This definition of equality underscores the distinct nature of the real and imaginary components, emphasizing their independent roles in defining a complex number.
In the realm of complex number simplification, certain keywords serve as guiding beacons, illuminating the path towards efficient solutions. These keywords, often representing fundamental concepts or properties, empower us to navigate complex expressions with confidence. Let's delve into the significance of some key keywords in the context of simplifying (1 + 7i)(3 - i).
Distributive Property: The Cornerstone of Expansion
The distributive property stands as a fundamental principle in algebra, serving as the cornerstone for expanding expressions involving parentheses. In the context of complex number multiplication, the distributive property allows us to systematically multiply each term in one complex number by each term in the other, effectively breaking down the expression into manageable components. Without the distributive property, simplifying (1 + 7i)(3 - i) would be an arduous task, fraught with potential errors.
Imaginary Unit (i): The Gateway to Complex Numbers
The imaginary unit, denoted by i, is the defining characteristic of complex numbers, serving as the gateway to this extended number system. The property i² = -1 is paramount in simplifying complex expressions, enabling us to eliminate the imaginary unit from certain terms and express the result in the standard form a + bi. In simplifying (1 + 7i)(3 - i), the substitution of i² with -1 is a crucial step, without which the expression would remain in an unsimplified form.
Standard Form (a + bi): The Destination of Simplification
The standard form of a complex number, a + bi, serves as the ultimate destination in the simplification process. Expressing a complex number in this form provides a clear representation of its real and imaginary components, facilitating further operations and comparisons. The goal of simplifying (1 + 7i)(3 - i) is to transform the expression into this standard form, allowing us to readily identify its real and imaginary parts.
Simplifying expressions involving complex numbers, such as (1 + 7i)(3 - i), requires a firm grasp of complex number arithmetic, particularly the distributive property and the properties of the imaginary unit. By meticulously applying these principles, we can transform complex expressions into the standard form a + bi, unlocking their hidden structure and paving the way for further mathematical explorations. The journey through complex numbers is a testament to the beauty and elegance of mathematics, a realm where imagination and logic intertwine to reveal profound truths.
